RESEARCH REPORT | Applying Mathematics …...Ismail (Otago Polytechnic) assisted with the industry case studies in civil engineering. In 2014, Dr Ken Louie (Waikato Institute of Technology)

Applying mathematics knowledge to industry orientated problems in classroom teaching Ziming Tom Qi, Otago Polytechnic

May 2015

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Author

Associate Professor Ziming Tom Qi

2013 Project team

Associate Professor Ziming Tom Qi (Project Leader & Principal Investigator), Richard Nyhof, Dr.

Matt King and Dr. Najif Ismail, Otago Polytechnic

2014 Project team

Associate Professor Ziming Tom Qi and Dr. Mark Harmer, Otago Polytechnic

Dr. Ken Louie and Debbie Hogan, Waikato Institute of Technology

Daphne Robson, Christchurch Polytechnic Institute of Technology

Frank Cook, Wellington Institute of Technology

ACKNOWLEDGEMENTS

This report is based on contributions from the project team as detailed below:

In 2013, Richard Nyhof (Otago Polytechnic) provided the question list to industry leaders. He

assisted with the Otago Polytechnic Ethics Application. He also assisted with the

implementation of the industry case studies, data collection and analysis. Dr Matt King (Otago

Polytechnic) assisted with the industry case studies in mechanical engineering and Dr Najif

Ismail (Otago Polytechnic) assisted with the industry case studies in civil engineering.

In 2014, Dr Ken Louie (Waikato Institute of Technology) provided the formative test that was

implemented at the four Institutes of Technology and Polytechnics included in this project. He

assisted with the implementation of the industry case studies at Waikato Institute of Technology

and the data collection. He also assisted with gaining ethics approval from Waikato Institute of

Technology. Debbie Hogan assisted with gaining ethics approval from Waikato Institute of

Technology.

In 2014, Daphne Robson (Christchurch Polytechnic Institute of Technology) assisted with the

implementation of the industry case studies at Christchurch Polytechnic Institute of Technology,

data collection and data analysis. She also assisted with gaining ethics approval from

Christchurch Polytechnic Institute of Technology.

In 2014, Frank Cook (Wellington Institute of Technology) assisted with the implementation of the

industry case studies at Wellington Institute of Technology and data collection. He also assisted

with gaining ethics approval from Wellington Institute of Technology.

In 2014, Dr Mark Harmer (Otago Polytechnic) assisted with the implementation of the industry

case studies at Otago Polytechnic and data collection.

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I would also like to take this opportunity to thank Otago Polytechnic, Waikato Institute of

Technology, Wellington Institute of Technology, and Christchurch Polytechnic Institute of

Technology for supporting this study. Special acknowledgements are made to:

Jenny Aimers, Research Coordinator, Otago Polytechnic; Alistair Regan, Research and

Enterprise Director, Otago Polytechnic; Phil Ker, Chief Executive, Otago Polytechnic; John

Findlay, Head of School of Architecture Building and Engineering, Otago Polytechnic.

I would particularly like to express my thanks and acknowledge our debt to all those

organisations and individuals who have contributed to the preparation of this report. I hope

contributors will recognize that I have done my best to accurately reflect the variety of views and

the wealth of information which were so generously provided.

Finally, I would like to acknowledge Bridget O'Regan, Southern Hub Regional Manager, Ako

Aotearoa Southern Hub, for her support and unfailingly professional yet friendly advice. It is

much appreciated.

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Contents

ACKNOWLEDGEMENTS ............................................................................................................ i

EXECUTIVE SUMMARY ........................................................................................................... 2

INTRODUCTION ....................................................................................................................... 4

PROJECT AIMS ....................................................................................................................... 5

LITERATURE REVIEW .............................................................................................................. 5

METHODOLOGY ....................................................................................................................... 6

RESULTS AND FINDINGS ........................................................................................................ 7

Summary of findings ............................................................................................................... 7

Analysis of Stage 1 results ..................................................................................................... 7

Survey of engineering industry leaders ............................................................................... 7

Teaching using industry samples ........................................................................................ 7

Nature of the industry oriented questions ...........................................................................15

Survey of mathematics lecturers ........................................................................................16

CONCLUSION AND RECOMMENDATIONS ............................................................................17

REFERENCES .........................................................................................................................18

Appendix 1: Question list for focus group with industry leaders .................................................20

Appendix 2: Industry sample for civil engineering ......................................................................30

Appendix 3: Industry sample for mechanical engineering ..........................................................32

Appendix 4: 2014 Formative test trialed at the four Metro Group ITPs ......................................33

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EXECUTIVE SUMMARY

This project aimed to identify and test an industry oriented maths approach within four Institutes

of Technology and Polytechnics’ Bachelor of Engineering Technology (BEngTech) programmes.

The research question was: can BEngTech students apply the purely theoretical maths taught in

the current course to industry related maths problems?

The project was completed in two phases. Phase 1 was conducted by the research team at

Otago Polytechnic where 22 students were registered to the maths class and 16 of these

students attended the test for the research. Real life industry related maths problems were

collected from potential employers of BEngTech students and surveyed for maths knowledge. A

workshop was offered to the first year BEngTech maths students on industry related maths

problems in 2013. A comparison of the performance of students in theoretical maths problems

was undertaken on the maths results from the 2012 and 2013 student cohorts.

Phase 2 was conducted in 2014. The research methodology was redesigned to test a different

cohort of first year students at four metro Institutes of Technology and Polytechnics: Otago

Polytechnic, Waikato Institute of Technology, Wellington Institute of Technology, and

Christchurch Polytechnic Institute of Technology. After the students had completed the

theoretical maths course, they were tested to see if they could apply this maths knowledge to

solve industry related problems. These students did not undergo a workshop to explain the

industry related maths problem concept as detailed in Phase 1. Seventy six students from these

four metro Institutes of Technology and Polytechnics registered in the maths class and 36 of

these students completed the test for the research.

The findings of this project were:

1. Most first year BEngTech students who completed the test had difficulty with applying

their maths knowledge to industry oriented problems.

2. The engineering industry’s need for mathematical knowledge in the workplace varied

between the different majors, i.e. mechanical, civil or electrical.

The recommendations of this project are:

1. That further research is carried out in order to determine whether students need to be

taught additional skills for them to be able to apply theoretical knowledge in industry

contexts. One example is the development of scenario or exemplar questions.

2. That the same 2014 cohort is tested in their final year (2016) with the industry applied

assessment to see if their ability to apply their maths knowledge has been improved by

their exposure to the entire curriculum, not just maths.

3. That the Stage 1 half day workshop is replicated for all first year maths students in 2015

across the four ITPs following which these students be tested with both the standard

assessment and the industry applied assessment.

4. That a more detailed research project be formulated that includes interviews with

employers to investigate their specific needs with regard to the application of

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mathematical knowledge in the industry and continues to build a library of real world

problems.

5. That the research extends to a Phase Three which would explore the implementation of

industry project based learning as an alternative to the current method used of

theoretical based teaching.

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INTRODUCTION

The Bachelor of Engineering Technology (BEngTech) degree has been developed as a joint

venture between industry and the six largest New Zealand Institutes of Technology and

Polytechnics (ITPs): Unitec Institute of Technology (Unitec), Manukau Institute of Technology

(MIT), Wellington Institute of Technology (WelTec), Christchurch Polytechnic Institute of

Technology (CPIT), Waikato Institute of Technology (Wintec) and Otago Polytechnic (OP) who

form the metropolitan group of ITPs (Metro Group). This degree is provisionally accredited by

Institution of Professional Engineers New Zealand (IPENZ) and is recognised as meeting the

initial academic requirements for Engineering Technologists, as defined in the Sydney Accord –

an agreement developed for engineering technologists. Currently the programme has three

majors: Civil, Electrical and Mechanical Engineering.

New Zealand has a high demand for an increased number of Engineering Technologists. As this

new degree has an industry oriented commitment it is important that a new teaching and

learning methodology is developed in conjunction with industry leaders. The mathematics

component is the key to enhancing the capability of students to perform well in other

engineering courses within the programme as it is central to engineering practice (Paas et al.,

2004; Hawera & Taylor, 2007 and Cardella, 2010).

Engineering mathematics, which includes a branch of applied mathematics, is taught in the first

year of the BEngTech as a compulsory course. Thereafter students are not required to enrol in

a mathematics course again and have no chance to practise most of the teaching content learnt

from Engineering Mathematics.

Mathematics is an integral part of engineering education and is currently being taught in a

similar way to that of similar science-based degrees, i.e. as a pure theoretical subject. The

student profile of BEngTech students is however one where students are seeking a professional

or applied qualification rather than an academic pathway to further study. This leads to the

possibility that the BEngTech students will understand maths concepts better if they are taught

in an applied manner instead of as a purely theoretical subject.

This project contributes to the development of knowledge around industrial subject oriented

teaching and learning strategies and as such responds to the current government desire for

preparing students for work.

The Kaitohutohu at OP assisted with consultation for the project to gain ethical approval in 2013,

and then extended ethical approval in 2014. The project has been granted ethical approval at

the three other ITPs involved in this project: Wintec, WelTec, and CPIT.

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PROJECT AIMS AND OBJECTIVE

This project aimed to explore whether an industry oriented mathematics teaching strategy would

improve the mathematical achievement of learners in the Metro Group BEngTech programme.

This is important to the learner as achievement in mathematics is integral to other aspects of the

engineering programme. Understanding the mathematical underpinning of engineering

processes is a basic building block of all engineering disciplines, although the ways these are

applied to the different engineering specialties does differ. The research hypothesis was: the

needs of industry may require a shift from the teaching of generic or purely theoretical

mathematics to the industry specific mathematics courses in Civil, Electrical and Mechanical

Engineering.

LITERATURE REVIEW

The mathematics component in a BEngTech programme is critical to enhancing the capability of

students to perform well in other courses (Cardella, 2010; Gainsburg, 2006, and Engelbrecht et

al., 2012). There is a plethora of research to support an industry oriented approach to teaching

and learning mathematics in electronic and electrical engineering (Qi & Cannan, 2005, and Qi &

Cannan, 2006b) and undergraduate software engineering courses (Alsmadi & Hanandeh, 2011,

and Su et al., 2007) but there is currently a lack of literature with regard to civil and mechanical

engineering.

In traditional technical teaching methodologies the conventional educational pathway is to build

foundation learning through subject based teaching math, physics and science independently

(Bachelor of Engineering, 2012; Bachelor of Engineering Degree structure, 2012, and

Engineering technologist, 2012). Subjects based on the knowledge required for the discipline

usually follow on from this. The problem with this traditional methodology of learning is that

there is no close relationship with industry requirements. Students may well graduate with no

industry oriented learning experience prior to their first job. Industry oriented methodology is

learning from an industry perspective (Industry-oriented education, 2012, and Qi & Cannan,

2004).

As an example, the course of Electronics Technology in the BEngTech at Unitec Institute of

Technology was directly linked to industry and the focus was on an industry oriented product

such as a Switch-mode power supply (Qi & Cannan, 2005, and Qi & Cannan, 2006b). The focus

for learning was product design, application and operation of electronic components and

circuitry. Initially students received a demonstration and the product enclosure was opened to

investigate inside. The internal components forming the topics for study included the mechanical

design for the enclosure, electronic design including the PCB (Printed Circuit Board) and

embedded software design. An industry oriented product was used to simulate industry

conditions where students will gain invaluable insight into design technology, operational

procedure and programming techniques (Qi & Cannan, 2005). All foundation skills can be

taught within these studies and the students are well prepared to develop further knowledge and

skills (Qi & Cannan, 2007a) in their final year through cooperative education with industry.

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Under this model mathematics was totally integrated into the compulsory technical courses

rather than as a standalone course.

This approach can be applied to a traditional engineering undergraduate programme (Qi,

2008b), an industry oriented and multi-discipline undergraduate degree (Qi & Cannan, 2007b)

and post-graduate programmes (Qi, 2008a, and Qi, 2009). For example, a “bridging”

technology course was designed to enable a Bachelor of Design graduate in Unitec to enter a

Master of Design programme in Unitec (Qi, 2008a, and Qi, 2009). This teaching approach

requires a change in role for the lecturer. In industry orientated education the lecturers needed

to build their industry background. The academic staff were encouraged to join the student

industry projects as supervisors to improve their industry background, while industry staff were

invited to teach as guest or part-time lecturers (Qi & Cannan, 2007c).

The researchers were aware that for some learners (Māori), mathematics and how it is offered

outside of a context is viewed by some as a subject that promotes the values of the dominant

culture (Hawera & Taylor, 2007). The Māori concept of ‘Ako’ encompasses learning and

teaching as a process intertwined with concepts of mōhio (knowledge) and māramataka

(understanding) which ultimately results in mātauraka (wisdom). This synthesises people, ideas

and the environment as part of a greater whole; “Education is considered to be a holistic

enterprise, so mathematics should be integrated with everyday life” (Hawera & Taylor, 2007).

Hawera & Taylor’s (2007) study of Māori school students found that students had difficulty

applying maths to everyday life, i.e. placing maths learning within a context that has meaning for

the learner/s. This would suggest that a more applied approach to maths teaching would also

benefit Māori learners.

METHODOLOGY

The research was undertaken as action research in two phases during 2013 (phase 1) and

2014 (phase 2). The first phase was undertaken in 2013 by the project team from OP. An

industry survey (20 requests were sent to engineering managers and six of them were

completed and returned) was conducted to determine the aspects of the current maths course

most needed in industry. The full questionnaire can be found in Appendix 1. Further interaction

with industry groups obtained authentic and appropriate case studies to use as teaching

examples. Individual interviews were undertaken with the principal lecturers in the three

engineering majors, Electrical, Civil and Mechanical, within the BEngTech programme at OP.

Some industry examples of maths application from the three majors were selected during

discussions with the researchers and industry people. These samples were used in a student

workshop in addition to the standard curriculum from previous years. The students were

assessed in the same way as the previous year cohorts, using theoretical based questions. A

comparison was also made between the 2012 and 2013 exam results and pass rates at OP.

These students gave informed consent and qualitative feedback.

In phase 2 the method was amended with the involvement of the three other metro ITPs: Wintec,

Weltec, and CPIT. Initially it was planned to gather additional data from other metro ITPs (such

7

as assignment results of the previous two years cohorts) to identify the pass rate and the grade

status of students in each major, as well as to determine their base level of understanding to

use as comparative data with phase 1 research. However, following discussion, this was

changed to provide a formative test in the existing 2014 teaching plan for the maths course

within the Metro Group BEng Tech programme. This was additional, without modifying any

teaching content and learning outcomes in the class.

The formative test was developed in two parts: The first question was generic and involved

finding the determinant and inverse of a matrix related to a system of simultaneous equations.

The second question allowed the students to choose from a) civil, b) electrical, or c) mechanical

areas but all involved setting up and solving a system of simultaneous equations using a matrix

method. The results of the first question could then be compared with the second. Appendix 4

details the formative test that was trialed on the students at the four Metro ITPs.

RESULTS AND FINDINGS

Summary of findings

The study produced four broad findings.

Phase 1: The OP team found:

The engineering industry requirement for mathematical knowledge in the workplace varied

between the different majors, i.e. mechanical, civil or electrical.

Phase 2: The collaborative project found:

Students from all the current engineering math classes in the research programme had difficulty

applying their knowledge to solve an industry application, even if their mathematical knowledge

was rated satisfactory in the theoretical application of the test.

These broad findings are discussed in more detail below.

Analysis of Phase1 results

Survey of engineering industry leaders

Phase 1 surveyed engineering industry leaders using the full questionnaire shown in Appendix 1. As shown in Appendix 5, it was found that while the BEng Tech taught a generic maths course, depending on the industry major, the engineering industry had differing needs for maths knowledge.

Teaching using industry samples

After some industry examples of maths application from the three majors (shown in Appendices

2 and 3) were delivered in a workshop in addition to the standard curriculum from previous

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years, the students were assessed in the same way as the previous year cohorts, using

theoretical based questions. A comparison was made between the 2012 and 2013 exam

results and pass rates at OP. The detailed analysis of the student results in the standard

theoretical tests is shown in Appendix 3. OP had civil and mechanical students for this research.

Our conjecture was that the civil major students in 2013 with the benefit of the workshop would

do better than the 2012 students on the Matrices question while the mechanical major students

in 2013 would do better than the 2012 students on the Series question.

Base mean

The means for each question was calculated for the different years and streams. As shown in

Figures 1 and 2, there is a very small change between the marks obtained in 2012 and 2013 for

each stream. However they are the opposite of what was conjectured. From looking at the

graphs we can see that:

Civil stream students dropped 3% on matrices question

Mechanical stream students gained 1% on matrices question

Civil stream students gained 2% on series question

Mechanical stream students dropped 1% on series question.

These are very small changes in percentage so no immediate significance is seen.

Figure 1 Civil engineering sample

2012 2013

Civil 78 75

Mechanical 71 72

0

20

40

60

80

100

Exam

Qu

esti

on

Per

cen

tage

Matrices (Civil Industry Example)

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Figure 2 Mechanical engineering sample

Statistical testing

A test was run with a 95% significance level to examine the change in the Phase 1 data. The

same test was run on each of the questions for both streams to check for any significance

across the board.

F-TEST

An F-test is used first to determine the variance of the data collected as showed in Table 3-6.

Table 1 F-test at Civil Stream (Matrices)

Civil Stream (Matrices)

2012 2013

Mean 77.5568182 75.37879

Variance 440.433146 339.7039

Observations 8 6

Table 2 F-test at Civil Stream (Series)

Civil Stream (Series)

2012 2013

Mean 48.61111 51.38889

Variance 359.3474 869.5988

Observations 8 6

2012 2013

Civil 49 51

Mechanical 62 61

0

20

40

60

80

100

Exam

Qu

esti

on

Per

cen

tage

Series (Mechanical Industry Example)

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Table 3 F-test at Mechanical Stream (Matrices)

Mechanical Stream (Matrices)

2012 2013

Mean 70.9090909 72.34848

Variance 813.59045 673.2094

Observations 10 12

Table 4 F-test at Mechanical Stream (Matrices)

Mechanical Stream (Series)

2012 2013

Mean 62.22222 61.11111

Variance 375.8573 735.1291

Observations 10 12

The variances are all greatly different for each data set. This determines that any t-tests used

will need to be performed with unequal variances in mind.

T-test

A t-test was performed on each selection of data to determine if it is of any statistical importance.

Null hypothesis: that the difference in mean between 2013 and 2012 is equal.

Alternative hypothesis: that the difference in mean for 2013 is greater than in 2012.

Significance level: 95%

A t-test was performed to check if there was any statistical evidence that the marks in 2013

were significantly greater when compared with the 2012 marks. A two tailed test was performed

to check if there is any statistical evidence that the marks in 2013 were significantly different

when compared with the 2012 marks.

Table 5 T-test at Civil Stream (Matrices)

Civil Stream (Matrices)

t-test: Two-sample Assuming Unequal Variances

Hypothesized Mean Difference 0

df 12

t Stat 0.20610704

P(T<=t) one-tail 0.42008135

t Critical one-tail 1.78228756

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P(T<=t) two-tail 0.8401627

t Critical two-tail 2.17881283

Table 6 T-test at Civil Stream (Series)

Civil Stream (Series)

t-test: Two-sample assuming Unequal Variances


df 8

t Stat -0.2015999

P(T<=t) one-tail 0.42263064


P(T<=t) two-tail 0.84526127


Table 7 T-test at Mechanical Stream (Matrices)

Mechanical Stream (Matrices)

t-test: Two-Sample Assuming Unequal Variances


df 18

t Stat -0.1227699

P(T<=t) one-tail 0.45182486


P(T<=t) two-tail 0.90364972


Table 8 T-test at Mechanical Stream (Series)

Mechanical Stream (Series)

t-test: Two-sample Assuming Unequal Variances


df 20

t Stat 0.11175755

P(T<=t) one-tail 0.45606483


P(T<=t) two-tail 0.91212967


In Tables 7 to 10 we can clearly see that the Critical t value is much higher than the t stat. This

means that there is no statistical importance between the marks from 2012 and 2013 in the

algebra exam regarding the matrices and series questions.

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Our conjecture was that the civil major students in 2013 with the benefit of the workshop would

do better than the 2012 students on the Matrices question while the mechanical major students

in 2013 would do better than the 2012 students on the Series question.

The t-test results show that the conjecture has no statistical evidence to back it up, and we must

instead accept the null hypothesis stated, namely that the difference in mean between 2013 and

2012 is equal. Below is a graph showing the overall exam mean for the combined streams in

both years: It is interesting to note that the difference in the overall means for both years’ exams

results is only 0.2%. This shows great consistency and is a good indicator that we could expect

the whole exam to have been consistent with no major change in exam marks as showed in

Figure 3.

Figure 3

Analysis of Phase 2 results

Students were offered two questions. Question 1 (Q1) was a standard findings theoretical maths

question and question 2 (Q2) asked students to apply their theoretical knowledge to an industry-

based question. The results are separated and analysed by student major of mechanical, civil,

or electrical engineering across the four ITPs. The results were compared after all tests were

completed so there is no way of matching the test results to specific students, and any results of

the study will only report the anonymized data.

The collated data from the four participating ITPs is shown in Table 11. Sixty one per cent of

students completed both Q1 and Q2.

1. Students did a lot better on Q1 than Q2 scoring an average of 70% on Q1 and 35% on

Q2 and the difference is statistically significant (p<0.00000001).

2012 2013

Combined 59.3 59.1

0.0

20.0

40.0

60.0

80.0

100.0

Exam

To

tal P

erce

nta

ge Overall Exam Marks

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2. If we only consider those students who attempted Q2, it can still be seen that they did

better on Q1 than Q2 although the difference is less marked. Their average scores were

79% for Q1 and 59% for Q2 and once again the difference is statistically significant

(p<0.001).

Table 9 Collated Data from the four institutes

Student category Number

Q1 mean

score

(%)

Q2 mean

score

(%)

All 76 70 35

Attempted BOTH questions 46 79 59

Attempted Q1 only 30 56

Mechanical attempted both

questions 18 73 61

Electrical attempted both

questions 8 83 67

Civil attempted both

questions 20 82 54

3. Table 12 shows the correlation coefficients of the comparison. There is little correlation

between Q1 and Q2 for any of the groups of students as the correlation coefficients are

Mechanical 0.48, Electrical 0.22, and Civil 0.13. This poor correlation can be seen

graphically in Figures 4 to 6.

Table 10 Correlation coefficient, r, between Q1 and Q2

r (Q1 and Q2)

Mechanical 0.48

Electrical 0.22

Civil 0.13

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Figure 4 Student scores - Civil Q2 vs Q1

Figure 5 Student scores - Electrical Q2 vs Q1

0

10

20

30

40

50

60

70

80

90

100

0 20 40 60 80 100

Q2

sco

res

Q1 scores

Civil

0

10

20

30

40

50

60

70

80

90

100

0 20 40 60 80 100

Q2

sco

res

Q1 scores

Electrical

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Figure 6 Student scores - Mechanical Q2 vs Q1

4. Many students did not appear to have enough confidence to apply skills shown in Q1 to the

industry-oriented application in Q2.

5. About half the students used the inverse matrix method for Q1 part (d), and about half used

the Gaussian method, although parts (a) to (c) guided them towards the inverse matrix

method. A few students only used their calculator, although full working had been requested.

In Q2, most students used the Gaussian method. This method appears to be the most

popular of the three matrix methods taught in the course.

Nature of the industry oriented questions

For industry oriented questions (Q2), students needed skills that were additional to those

needed for the theoretical questions (Q1) where the equations were given in standard textbook

form and early parts of the question guided students towards a particular method. In Q2

however, students needed to rewrite the equations into standard textbook form and received no

guidance towards a method. Thus they needed to read, understand, and interpret both words

and diagrams.

In addition, the three options for Q2 had other characteristics that made them more difficult to

analyse than Q1. These are shown in Table 13.

0

10

20

30

40

50

60

70

80

90

100

0 20 40 60 80 100

Q2

sco

res

Q1 scores

Mechanical

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Table 11 Characteristics of Q2

Q2a

Civil

Needed to show where the equations came from

Needed to deal with zero coefficients

One equation needed rearranging

Q2b

Electrical

Needed to substitute values

Needed to know Kirchoff’s Laws

Q2c

Mechanical

There were four unknowns

Needed to deal with zero coefficients

Needed to substitute values and these were not given until part (ii) of the question

Discussions with mathematics lecturers

As part of the action research process we discussed with mathematics lecturers whether they

would expect to teach mathematics and whether their students would be taught how to apply

this to industry applications in their engineering courses. We found that

There are a number of additional skills needed to “use mathematics” and it appears that

these skills, some of which are identified in these discussions, may not be being taught

in either the mathematics or the engineering courses at year 1 of BEngTech.

The mathematics lecturers expressed a willingness to teach these industry based

problems. However, although they have background of basic engineering, they felt they

are not necessarily confident or competent to teach engineering mathematical

application. Similarly, engineering teachers understand the mathematics they use but

may not feel confident about teaching that aspect.

No evidence is available yet to show whether students would develop some of the

addition skills needed to use maths as they continue in years 2 or 3 of their BEngTech

programme.

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CONCLUSION AND RECOMMENDATIONS

At phase 1, the engineering industry showed support for industry oriented teaching and learning.

Their input identified however that there were some different needs for maths knowledge and

variations in application between the different majors.

At phase 2, students in maths classes lack the skills to apply their mathematical knowledge to

industry oriented questions. However, as the students tested were in their first year of study,

there may be other variables that improve competence as students move through their entire

course of study.

This raises further questions such as, what role does industry oriented teaching and learning

have on a student’s ability to apply mathematical knowledge? Would their achievement improve

once they are taught industry based courses? We also noted whether traditional mathematics

lecturers would expect to teach the mathematics and assume that students would be taught

how to apply this to industry applications in their engineering courses. More research is needed

to determine whether the additional skills needed by students to apply maths is developed as

students progress in years 2 and 3 and whether it needs to be taught in either the mathematics

or the engineering courses.

Based on reflections on this project, it is recommended that this project be extended to phase 3,

which could address the following:

Test the same 2014 cohort of students in their final year (2016) with the industry applied

assessment to see if their ability to apply their maths knowledge has been improved by

their exposure to the entire curriculum, not just maths.

Replicate the phase 1 half day workshop for all first year maths students in 2015 across

the four study ITPs and test these students with both the standard assessment and the

industry applied assessment.

Undertake more detailed research with employers about their specific needs with regard

to the application of mathematical knowledge and continue to build a library of real world

problems.

Taking into account the applied nature of the BEngTech training this project could be

extended to phase 3 to trial the implementation of project based learning as an

alternative to the current method used of theoretical based teaching.

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20

Appendix 1: Question list for focus group with industry leaders

Question list for focus group with industry leaders

The purpose of this research is to:

identify math concepts used in your field

For any topic below which you would apply “our engineers must know”, could you please

provide an example of where this concept is used in your field.

Tick below box to identify your specialist area

Civil Engineering

Electrical Engineering

Mechanical Engineering

ALGEBRA

1 FUNCTIONS AND COORDINATE SYSTEMS

1.1 Define and graph a relation and its inverse where the relation is:

1.1.1 simple polynomial eg. y = kxa

1.1.2 exponential

1.1.3 circular

1.1.4 hyperbolic

Our engineers must know

Our engineers needn’t know

not sure

1.2 Convert among common coordinate systems



not sure

1.3 Graph plane curves in polar co-ordinates



not sure

2. VECTOR ALGEBRA

21

2.1 Describe a 3 space vector as an ordered triple and in terms of the unit vectors i, j,

k (include discussion of 2 space)



not sure

2.2 Perform vector addition, subtraction and multiplication by scalar quantities



not sure

2.3 Calculate the magnitude and the directed unit vector corresponding to a vector v



not sure

2.4 Find the vector normal and vector tangent to a simple curve at a specified point

Find: (a) the vector between 2 points A, B in 3 space

(b) the distance between the points A,B



not sure

2.5 Define the scalar product and describe its geometric significance in terms of projections



not sure

2.6 Apply the scalar product to simple physical problems



not sure

2.7 Define and find direction cosines



not sure

2.8 Define and evaluate determinants (up to order 3)



not sure

2.9 Define the cross product and describe its geometric properties



not sure

2.10 Verify the distributive law for the cross product



not sure

22

2.11 Apply the cross product to simple physical problems



not sure

2.13 State the equation of a plane



not sure

2.14 Find a vector N, normal to a plane



not sure

2.15 Describe the algebraic and geometric properties of the triple scalar product



not sure

2.16 Apply vectors to engineering applications



not sure

3 LINEAR ALGEBRA

3.1 Apply the rules for matrix addition, subtraction and scalar multiplication



not sure

3.2 Apply the rule for matrix multiplication



not sure

3.3 State the additive and multiplicative matrix identities



not sure

3.4 Investigate the commutative, associative and distributive properties of matrices



not sure

3.5 Apply the rules governing elementary row operations to obtain an inverse matrix


23


not sure

3.6 Show that a consistent set of linear equations may be represented in matrix form and hence find a solution set (using a variety of standard methods)



not sure

3.7 Understand the term 'row echelon form'



not sure

3.8 Find an inverse matrix by cofactors



not sure

3.9 Determine the Eigen values of a matrix



not sure

3.10 Determine the Eigen vectors for a matrix



not sure

3.11 Apply Eigen vectors to the linear transformation of Cartesian coordinate systems



not sure

3.12 Use matrices to solve engineering problems



not sure

4 COMPLEX ALGEBRA

4.1 Convert between Cartesian and polar forms



24

not sure

4.2 Represent these forms on a drawing of the complex plane



not sure

4.3 Apply Euler's formula



not sure

4.4 Derive the fundamental identity 01 je



not sure

4.5 Verify the complex links between sinz, cosz, sinhz and coshz



not sure

4.6 Deduce and use Osborne's Rule Our engineers must know


not sure

4.7 Define and evaluate the following functions: sinz, cosz, lnz, sinhz, coshz, zn and

combinations of these as appropriate



not sure

4.8 Find roots of complex numbers



not sure

25

4.9 Show the importance of complex numbers in professional engineering calculations



not sure

5 SERIES

5.1 Understand the term “limit” and the notion of convergence



not sure

5.2 Use L’Hopital’s Rule



not sure

5.3 Write a Maclaurin Series for transcendental functions



not sure

CALCULUS

1 DIFFERENTIAL CALCULUS

1.1 Review of differentiation concepts and methods



not sure

1.2 Perform implicit differentiation



not sure

1.3 Recognise a composite function and state the Chain Rule for its derivative



not sure

1.4 Understand the terms concavity, critical point, inflexion, continuity, increasing and decreasing


26


not sure

1.5 Define and apply the operator



not sure

1.6 Apply the Chain Rule for functions of more than one variable



not sure

1.7 Use partial derivatives to explore 1.4



not sure

1.8 Apply partial differentiation to solve practical engineering problems



not sure

2 INTEGRAL CALCULUS

2.1 Review of integration concepts and methods



not sure

2.2 Find an integral by expansion to partial fractions



not sure

2.3 Find an integral by trigonometric or hyperbolic substitution



not sure

2.4 Apply integration techniques to find the length of a plane curve



not sure

2.5 Apply integration techniques to find the area of a surface of revolution



not sure

2.6 Apply integration techniques to find the volume of a solid of revolution



27

not sure

2.7 Apply the method of integration by parts to integrate products of functions (excluding reduction formulae)



not sure

2.8 Evaluate and use double integrals



not sure

2.9 Apply integration to solve practical engineering problems



not sure

3 DIFFERENTIAL EQUATIONS

3.1 Define the terms: 3.1.1 differential equation

3.1.2 order

3.1.3 degree

3.1.4 initial condition

3.1.5 boundary condition

3.1.6 general solution



not sure

3.2 Distinguish first and second order Linear D.Es



not sure

3.3 Recognise that the general solution of a D.E. describes a family of curves and that the particular solution describes a unique curve



not sure

3.4 Solve a first order linear DE by:

3.4.1 direct integration

28

3.4.2 separation of variables 3.4.3 integrating factor 3.4.4 substituting y = bx for homogeneous DEs



not sure

3.5 Solve a homogeneous second order D.E. of the form ay" + by' + cy = 0

for all real a,b,c



not sure

3.6 Apply the above techniques, where appropriate, to find the particular solution for

engineering problems.



not sure

4 NUMERICAL METHODS

4.1 Construct Newton/Gregory difference tables



not sure

4.2 Find an interpolating polynomial from data



not sure

4.3 Use an interpolating polynomial to determine the value of slope at a particular point.



not sure

4.4 Use the trapezium rule.



29

not sure

4.5 Use Simpson's rule.



not sure

4.6 Apply numerical techniques to solve problems in Calculus



not sure

4.7 Use an iterative method (such as Euler’s method) to solve simple first

order differential equations.



not sure

4.8 Know that a numerical process has an associated error bound and that such a bound should be evaluated and along presented with the numerical answer itself



not sure

30

Appendix 2: Industry sample for civil engineering

The following sample was created based on the response from engineers in the relevant specialist “must know” as shown in Appendix 1.

CIVIL 1.1 Define and graph a relation and its inverse where the relation is:

Civil engineers would plot several parameters to reach substantial conclusions e.g., plot stress-

strain curves to get the dependable yield strength of steel, time and spectral acceleration to get

the design spectra etc.

Example 1: Readings from a typical steel test, in terms of force and displacement are given

below. You are required to draw a typical stress strain plot and find the yield strength (i.e., at the

first change of slope), elastic modulus (i.e., the slope of initial straight line) and ultimate strength

of the reinforcement steel (i.e., maximum stress before breakage).

[Note: Cross-sectional area of the bar (12 mm diameter bar) = 113.1 mm2 and Gauge

length = 100 mm]

Sr. No. Force (kN) Displacement (mm)

1 0.00 0.00 2 32.78 0.15 3 47.92 0.22 4 58.67 0.28 5 63.71 0.32 6 66.80 0.35 7 68.57 0.38 8 69.96 0.40 9 70.94 0.42 10 71.75 0.44 11 74.75 0.54 12 75.97 0.67 13 77.17 1.02 14 76.63 1.45 15 75.02 1.83 16 68.33 2.28

1.1.1 simple polynomial eg. y = kxa

1.1.2 exponential

1.1.3 circular

31

1.1.5 hyperbolic

1.2 Convert among common coordinate systems Knowledge of global and local Cartesian and Polar coordinate systems is crucial in surveying

and for field engineers to set out footprint plans. Additionally, structural modelling and drafting

requires good comprehension of coordinate systems.

Example 2: A typical site layout plan (for site boundary demarcation) is given below in global

coordinate system (northing and easting). You are required to work out the lengths of each side

and all internal angles from given co-ordinates to verify the site layout.

4.1 Convert between Cartesian and polar forms 4.3 Apply Euler's formula

4.4 Derive the fundamental identity 01 je

4.5 Verify the complex links between sinz, cosz, sinhz and coshz

4.6 Deduce and use Osborne's Rule 4.7 Define and evaluate the following functions: sinz, cosz, lnz, sinhz, coshz, zn and

combinations of these as appropriate

4.8 Find roots of complex numbers

32

4.9 Show the importance of complex numbers in professional

engineering calculationsIndustry sample for mechanical engineering

33

Appendix 4: 2014 Formative test trialed at the four Metro Group ITPs

Test Q1: Matrices and simultaneous equations

Students are required to complete below standard questions:

1) This question refers to the following system of equations: 𝑥 − 𝑦 + 3𝑧 = 3

2𝑥 + 𝑦 + 𝑧 = 7

−3𝑥 + 𝑦 + 4𝑧 = 9

a) Write the system of equations in the form Ax = b (1 mark)

b) Calculate det A, (determinant of A) (2 marks)

c) Find inv(A), (inverse of A) (2 marks)

d) Use any matrix method to solve the system of equations. Clearly show your process to obtain full marks. (5 marks)

34

Test Q2: Industry oriented questions

Students are encouraged to select one of the industry-oriented questions below.

a) A civil industry-oriented application: In an industrial process water flows through three tanks in succession as illustrated in the

figure. The tanks have unit cross-section and have heads (levels) of water 𝑥, 𝑦 and 𝑧

respectively. The rate of inflow into the first tank is 𝑢, the flowrate in the tube connecting

tanks 1 and 2 is 5(𝑥 − 𝑦), the flowrate in the tube connecting tanks 2 and 3 is 4(𝑦 − 𝑧) and

the rate of outflow from tank 3 is 6𝑧

Figure 7 Industrial process: water flows through three tanks

i. Show that the equations of the system in the steady flow situation are 𝑢 = 5𝑥 − 5𝑦

0 = 5𝑥 − 9𝑦 + 4𝑧

0 = 4𝑦 − 10𝑧

(2 marks)

ii. By solving this system of linear equations find 𝑥, 𝑦 and 𝑧. (5 marks)

5(𝑥 − 𝑦) 4(𝑦 − 𝑧) 6𝑧

35

b) An electrical industry-oriented application:

The figure illustrates an electrical network with mesh currents 𝐼1, 𝐼2 and 𝐼3 shown.

Figure 8 Electrical network with mesh currents

i. By applying Kirchhoff’s voltage law show that the matrix equation for 𝐼1, 𝐼2 and 𝐼3 is given

by

(

𝑅1 + 𝑅2 −𝑅1 −𝑅2

𝑅1 −𝑅1 0

𝑅2 0 −(𝑅2 + 𝑅3)) (

𝐼1

𝐼2

𝐼3

) = (𝐸1

𝐸2

𝐸3 − 𝐸2

)

(2 marks)

ii. Calculate 𝐼1, 𝐼2 and 𝐼3 given 𝐸1 = 5 V, 𝐸2 = 6 V and 𝐸3 = 12 V, 𝑅1 = 15 𝛺 , 𝑅2 = 5 Ω and 𝑅3 = 10 𝛺.

(5 marks)

36

c) A Mechanical industry-oriented application:

A cantilever beam bends under a uniform load 𝑤 per unit length and is subject to an axial

force 𝑃 at its free end. For small deflections a numerical approximation to the shape of the

beam is given by the set of equations

−𝑣𝑦1 + 𝑦2 = −𝑢

𝑦1 − 𝑣𝑦2 + 𝑦3 = −4𝑢

𝑦2 − 𝑣𝑦3 + 𝑦4 = −9𝑢

2𝑦3 − 𝑣𝑦4 = −16𝑢

These deflections are indicated in the figure below. The parameters 𝑢 and

𝑣 are related to the flexural rigidity, axial load and length of the beam.

Figure 9 A cantilever beam bends under a uniform load

i. Write the set of equations in matrix form (1 mark)

ii. Use any method you know to solve these equations when the parameter values are 𝑢 = 1 and

𝑣 = 3. (6 marks)

37

Appendix 5: Summary of the answers to question list for focus group

with industry leaders Questions Civil

Engineering Electrical Engineering

Mechanical Engineering

ALGEBRA

FUNCTIONS AND COORDINATE SYSTEMS

Define and graph a relation and its inverse 1,1 1,1 1,1

Convert among common coordinate systems 1,1 1,1 1,1

Graph plane curves in polar co-ordinates 2,1 1,1 3,2

VECTOR ALGEBRA

Describe a 3 space vector as an ordered triple and in terms of the unit vectors

1,1 1,1 1,2

Perform vector addition, subtraction and multiplication by scalar quantities

3,1 1,1 1,1

Calculate the magnitude and the directed unit vector corresponding to a vector

3,1 1,1 1,1

Find the vector normal and vector tangent to a simple curve at a specified point

1,3 3,1 1,2

Define the scalar product and describe its geometric significance in terms of projections

1,3 3,1 1,2

Apply the scalar product to simple physical problems 1,3 3,1 1,2

Define and find direction cosines 1,3 3,1 1,2

Define and evaluate determinants (up to order 3) 1,3 2,1 3,2

Define the cross product and describe its geometric properties 1,3 2,2 2,2

Verify the distributive law for the cross product 2,3 2,2 2,2

Apply the cross product to simple physical problems 2,3 2,2 2,2

State the equation of a plane 2,3 3,1 2,2

Find a vector N, normal to a plane 2,3 3,1 3,2

Describe the algebraic and geometric properties of the triple scalar product

1,3 3,2 3,2

Apply vectors to engineering applications 1,1 1,1 1,1

LINEAR ALGEBRA

Apply the rules for matrix addition, subtraction and scalar multiplication

1,3 2,1 3,2

Apply the rule for matrix multiplication 2,3 2,1 3,2

State the additive and multiplicative matrix identities 2,3 2,1 2,2

Investigate the commutative, associative and distributive properties of matrices

2,3 2,1 3,2

Apply the rules governing elementary row operations to obtain an inverse matrix

2,3 3,1 2,2

Show that a consistent set of linear equations may be represented in matrix form and hence find a solution set (using a variety of standard methods)

2,1 2,1

3,2

Understand the term 'row echelon form' 2,3 2,1 2,2

Find an inverse matrix by cofactors 2,3 2,3 3,2

Determine the Eigen values of a matrix 2,3 2,3 3,2

Determine the Eigen vectors for a matrix 2,3 2,3 3,2

Apply Eigen vectors to the linear transformation of Cartesian coordinate systems

2,3 2,3 3,2

Use matrices to solve engineering problems 2,1 2,1 1,2

COMPLEX ALGEBRA

Convert between Cartesian and polar forms 1,1 1,1 1,1

Represent these forms on a drawing of the complex plane 2,3 1,1 1,2

Apply Euler's formula 1,3 1,1 1,2

Derive the fundamental identity 1,3 3,1 2,2

Verify the complex links between sinz, cosz, sinhz and coshz 1,3 3,1 1,2

Deduce and use Osborne's Rule 1,3 3,3 2,2

Define and evaluate the following functions: sinz, cosz, lnz, 1,3 3,3 1,2

38

sinhz, coshz, zn and combinations of these as appropriate

Find roots of complex numbers 1,3 3,3 2,2

Show the importance of complex numbers in professional engineering calculations

1,3 1,1 2,1

SERIES

Understand the term “limit” and the notion of convergence 2,3 1,1 1,2

Use L’Hopital’s Rule 3,3 1,3 3,2

Write a Maclaurin Series for transcendental functions 2,3 3,3 3,2

CALCULUS

DIFFERENTIAL CALCULUS

Review of differentiation concepts and methods 1,3 2,1 1,1

Perform implicit differentiation 2,3 2,1 1,2

Recognise a composite function and state the Chain Rule for its derivative

2,3 2,1 1,2

Understand the terms concavity, critical point, inflexion, continuity, increasing and decreasing

1,3 2,3 1,2

Define and apply the operator 2,3 3,2 1,2

Apply the Chain Rule for functions of more than one variable 2,3 3,2 1,2

Use partial derivatives to explore 1.4 3,3 2,2 1,2

Apply partial differentiation to solve practical engineering problems

2,3 2,2 1,2

INTEGRAL CALCULUS

Review of integration concepts and methods 1,3 1,1 1,1

Find an integral by expansion to partial fractions 2,3 3,3 1,2

Find an integral by trigonometric or hyperbolic substitution 2,3 3,3

3,2

Apply integration techniques to find the length of a plane curve 2,3 3,3 1,2

Apply integration techniques to find the area of a surface of revolution

2,3 3,3 1,2

Apply integration techniques to find the volume of a solid of revolution

2,3 3,3 1,2

Apply the method of integration by parts to integrate products of functions (excluding reduction formulae)

2,3 1,3 1,2

Evaluate and use double integrals 2,3 2,3 1,2

Apply integration to solve practical engineering problems 2,3 1,1 1,2

DIFFERENTIAL EQUATIONS

Define the terms: differential equation, order, degree, initial condition, boundary condition, general solution

1,1 2,1 1,2

Distinguish first and second order Linear D.Es 1,1 2,1 1,2

Recognise that the general solution of a D.E. describes a family of curves and that the particular solution describes a unique curve

1,1 2,1 1,2

Solve a first order linear DE 1,1 23 1,2

Solve a homogeneous second order D.E 1,1 23 1,2

Apply the above techniques, where appropriate, to find the particular solution for engineering problems

1,1 2,1 1,2

NUMERICAL METHODS

Construct Newton/Gregory difference tables 1,3 2,3 2,2

Find an interpolating polynomial from data 1,3 3,1 1,2

Use an interpolating polynomial to determine the value of slope at a particular point

1,3 1,1 1,2

Use the trapezium rule 1,1 1,3 1,2

Use Simpson's rule 1,1 3,3 22

Apply numerical techniques to solve problems in Calculus 1,1 1,3 1,2

Use an iterative method (such as Euler’s method) to solve simple first order differential equations

1,1 3,1 1,2

Know that a numerical process has an associated error bound and that such a bound should be evaluated and along presented with the numerical answer itself

1,3 2,3 1,2

39

Notes:

As shown in Appendix 5, the results of the survey are tabulated, where “1” means “Our engineers must know”; “2” means “Our engineers needn’t know” while “3” means “not sure”.

Note that some results come with “1” and “2”at the same group, which are marked in red in Table 2, indicating the answers from the industry leaders in the same major have an appositive view.

This insufficient data shows significantly some teaching contents were selected as “Our engineers needn’t know” in a major at the same time. The results come with “1” and “1” mean the same view. Any result with “3” mean one of the views was ”not sure”.

Documents

RESEARCH REPORT | Applying Mathematics …...Ismail (Otago Polytechnic) assisted with the industry case studies in civil engineering. In 2014, Dr Ken Louie (Waikato Institute of Technology)